A representation of Jacobi functions

Abstract

Recently, the continuous Jacobi transform and its inverse are defined and studied in [1] and [2]. In the present work, the transform is used to derive a series representation for the Jacobi functions Pλ(α,β)(x), −½≤α, β≤½, α+β=0, and λ≥−½. The case α=β=0 yields a representation for the Legendre functions and has been dealt with in [3]. When λ is a positive integer n, the representation reduces to a single term, viz., the Jacobi polynomial of degree n.